How can I flatten a torus only on certain axes so that it looks more like a flattened band?
I assume you want a band with some thickness, like this:

There are obviously many ways to make such a shape. Here are a couple:
Scaling a torus:

Add torus

Scale on the Z (SZ)

Optionally enter edit mode (Tab) and scale along the normals with AltS



Circle with subsurf modifier:

Add a circle (ShiftA&gt; Mesh &gt; Circle)

Extrude (E) into a band shape

Add subsurf modifier (Ctrl1-5, where the number refers to the number of viewport subdivisions)

Optionally add loopcuts to flatten some sides of the band